If you wish to find a root of f(x)=0, consider re-writing the system as
g(x)=x. A fixed point is a value 
such that 
.
Fixed point iteration is an iterative procedure attempting to find a
fixed point. Start with an initial guess, called 
. The algorithm
is
How you choose to write the g matters in the iteration. For example,
written as 
, or as
. Start from (1.5,1.5). The
first iteration diverges, but the second converges.
The Jacobian matrix 
figures into whether the iteration process converges
or not.
The key issue is contained in the following
Theorem Let 
satisfy 
. Assume the component
functions of G are continuously differentiable in a neighborhood of ,
and 
. Then for an initial guess sufficiently
close, the fixed point iteration converges.
Notice that, for the second way or writing the iteration, 
,
and we do see convergence.
The key feature is to look at the difference 
.
A Taylor expansion leads to
where 
is the point at which the taylor coefficients are evaluated.
By taking norms, we see
